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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term

Runzhang Xu1*, Jie Liu1, Yi Niu2 and Shaohua Chen3

Author Affiliations

1 College of Science, Harbin Engineering University, Harbin, 150001, People’s Republic of China

2 College of Automation, Harbin Engineering University, Harbin, 150001, People’s Republic of China

3 Department of Mathematics, Cape Breton University, Sydney, B1P 6L2, Canada

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Boundary Value Problems 2013, 2013:42  doi:10.1186/1687-2770-2013-42

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/42


Received:24 April 2012
Accepted:11 February 2013
Published:1 March 2013

© 2013 Xu et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we study the initial-boundary value problem of the multidimensional viscoelasticity equation with nonlinear source term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M1">View MathML</a>. By using the potential well method, we first prove the global existence. Then we prove that when time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M2">View MathML</a>, the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.

1 Introduction

This paper considers the initial-boundary value problem (IBVP) of the multidimensional viscoelasticity equation with nonlinear source term

(1.1)

(1.2)

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M6">View MathML</a> is the unknown function with respect to the spacial variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M7">View MathML</a> and the time variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M8">View MathML</a> is a bounded domain.

The viscoelasticity equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M9">View MathML</a>

(1.4)

was suggested and studied by Greenberg et al.[1,2] from viscoelasticity mechanics in 1968. Under the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M10">View MathML</a> and higher smooth conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M11">View MathML</a> and the initial data, they obtained the global existence of classical solutions for the initial-boundary value problem of Eq. (1.4).

After that many authors [3-11] studied the global well-posedness of IBVP for Eq. (1.4). In [3-10] the global existence, uniqueness and stability of solution were studied thoroughly. And in [11] the blow up of solution was discussed. Furthermore, in [12-14] the global existence of solution for IBVP of some multidimensional viscoelasticity equation was considered. And in [11] the blow up of solution for IBVP of the multidimensional generalisation of Eq. (1.4) was proved. Recently, in [15] and [16], the IBVP of the multidimensional viscoelasticity equation with nonlinear damping and source terms

(1.5)

(1.6)

(1.7)

was studied, and by using the potential well method, the global existence of weak solution was proved under some assumptions on nonlinear functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M17">View MathML</a> and the initial data. But we do not know how the global solution behaves as the time goes to infinity, namely the asymptotic behaviour of problem (1.1)-(1.3) is still open up to now. In the present paper, we try to study this problem by the multiplier method [17-22].

The main purpose of present paper is to consider the asymptotic behaviour of solution for problem (1.1)-(1.3). Since in the proof of the asymptotic behaviour of solution the global existence theory is required, it is necessary to give the proof of global existence of solution for problem (1.1)-(1.3).

In this paper, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M16">View MathML</a> satisfy the following assumptions:

where the constants in (H1) and (H2) are all positive and satisfy

In this paper, we first give some definitions and lemmas (Section 2). Then we prove the global existence of solution (Section 3). Finally, we prove the asymptotic behaviour of solution (Section 4).

In this paper, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M22">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M25">View MathML</a>.

2 Preliminaries

In this section, we will give some definitions and prove some lemmas for problem (1.1)-(1.3).

For problem (1.1)-(1.3), we define

Remark 2.1 Note that the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M28">View MathML</a> in the present paper are different from those in [11] and [15]. The definitions given in this paper will be shown more natural and rational because they are a part of the total energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M29">View MathML</a>.

Lemma 2.2Let (H1) and (H2) hold. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M30">View MathML</a>

Then the following hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M31">View MathML</a>is increasing and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M32">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M33">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M34">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M33">View MathML</a>.

Proof This lemma follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M37">View MathML</a>. □

Lemma 2.3Let (H1) and (H2) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M38">View MathML</a>. Then the following hold:

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M39">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M40">View MathML</a>;

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M41">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M42">View MathML</a>;

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M43">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M44">View MathML</a>,

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M45">View MathML</a>

Proof

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M39">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M47">View MathML</a>

which gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M40">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M41">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M50">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M51">View MathML</a>

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M53">View MathML</a>, then by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M54">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M55">View MathML</a>

 □

Lemma 2.4Let (H1) and (H2) hold. Then the following holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M56">View MathML</a>

(2.1)

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M57">View MathML</a>, by Lemma 2.3, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M58">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M59">View MathML</a>

which gives (2.1). □

Now, for problem (1.1)-(1.3), we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M60">View MathML</a>

3 Global existence of solution

In this section, we prove the global existence of weak solution for problem (1.1)-(1.3).

Definition 3.1 We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M61">View MathML</a> a weak solution of problem (1.1)-(1.3) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M62">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M64">View MathML</a> satisfying

(i)

(ii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M66">View MathML</a>

Theorem 3.2Let (H1) and (H2) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M68">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M70">View MathML</a>. Then problem (1.1)-(1.3) admits a global weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M72">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M73">View MathML</a> be a system of base functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M74">View MathML</a>. Construct the approximate solutions of problem (1.1)-(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M75">View MathML</a>

satisfying

(3.1)

(3.2)

(3.3)

Multiplying (3.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M79">View MathML</a> and summing for s, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M80">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M81">View MathML</a>

(3.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M82">View MathML</a>

From (3.2) and (3.3), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M83">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M84">View MathML</a>. Hence, for sufficiently large n, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M85">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M86">View MathML</a>

(3.6)

On the other hand, since W is an open set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M74">View MathML</a>, Eq. (3.2) implies that for sufficiently large n, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M88">View MathML</a>. Next, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M89">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M90">View MathML</a> and sufficiently large n. If it is false, then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M91">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M92">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M94">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M95">View MathML</a>. So, by the definition of d, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M96">View MathML</a>, which contradicts (3.6).

From (3.6) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M97">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M98">View MathML</a>

and

which together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M89">View MathML</a> gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M101">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M102">View MathML</a>

(3.7)

From (3.7) we can get

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

Hence there exist u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M108">View MathML</a>, η and a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M109">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M110">View MathML</a> such that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M112">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M113">View MathML</a> weak-star, and a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M115">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M116">View MathML</a> weak-star and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M117">View MathML</a> weakly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M118">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M119">View MathML</a> weak-star, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M121">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M122">View MathML</a> weak-star, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M123">View MathML</a>.

Integrating (3.1) with respect to t, we have

(3.13)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M125">View MathML</a> in (3.13), we get

and

On the other hand, from (3.2) and (3.3), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M128">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M130">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M131">View MathML</a>. Therefore u is a global weak solution of problem (1.1)-(1.3). □

4 Asymptotic behaviour of solution

In this section, we prove the main conclusion of this paper - the asymptotic behaviour of solution for problem (1.1)-(1.3).

Lemma 4.1Let (H1) and (H2) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M68">View MathML</a>. Then, for the approximate solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M134">View MathML</a>of problem (1.1)-(1.3) constructed in the proof of Theorem 3.2, the following hold:

(i)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M135">View MathML</a>

(4.1)

(ii) Furthermore, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M136">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M70">View MathML</a>, then for sufficiently largen, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M138">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M139">View MathML</a>

(4.2)

Proof (i) Multiplying (3.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M140">View MathML</a> and summing for s, we get (4.1).

(ii) From

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M141">View MathML</a>

it follows that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M138">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M143">View MathML</a>

(4.3)

From (3.2), (3.3) and (4.3), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M144">View MathML</a> for sufficiently large n. Hence from (3.5) we have

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M146">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M147">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M148">View MathML</a>

which together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M149">View MathML</a> for sufficiently large n gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M150">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M151">View MathML</a>

Hence, by Lemma 2.3, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M152">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M153">View MathML</a>. So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M154">View MathML</a>

 □

Theorem 4.2Let (H1) and (H2) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M68">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M70">View MathML</a>. Then, for the global weak solutionugiven in Theorem 3.2, there exist positive constantsCandλsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M159">View MathML</a>

(4.4)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M110">View MathML</a> be the approximate solutions of problem (1.1)-(1.3) in the proof of Theorem 3.2, then (3.4) holds. Multiplying (3.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M161">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M162">View MathML</a>), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M163">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M164">View MathML</a>

(4.5)

From (H2), Lemma 2.2 and Lemma 4.1, we get

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M166">View MathML</a>

Hence we have

(4.6)

and

(4.7)

From

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M169">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M170">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M171">View MathML</a>

which together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M149">View MathML</a> for sufficiently large n gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M173">View MathML</a>

(4.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M174">View MathML</a>

(4.9)

From (4.8) and the Poincaré inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M175">View MathML</a>, it follows that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M176">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M177">View MathML</a>

(4.10)

From (4.5)-(4.10) it follows that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M178">View MathML</a> such that

(4.11)

Choose α such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M180">View MathML</a>

Then from (4.11) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M181">View MathML</a>

From this and the Gronwall inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M182">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M183">View MathML</a>

(4.12)

On the other hand, from (4.8) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M184">View MathML</a>

Hence, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M185">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M186">View MathML</a>

(4.13)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M109">View MathML</a> be the subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M110">View MathML</a> in the proof of Theorem 3.2. Then from (4.13) and(4.12), we obtain

which gives (4.4), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M191">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between all authors. RZ organised this paper and found the topic of this paper. He introduced this problem and suggested the methods and the outline of the proofs. JL finished the proof of the global existence and YN finished the long time behaviour part. SC discussed all the problems arising in the research and provided many good ideas for proving the problems. All authors have contributed to, seen and approved the manuscript.

Acknowledgements

We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11101102), Ph.D. Programs Foundation of the Ministry of Education of China (20102304120022), the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020), the Natural Science Foundation of Heilongjiang Province (A201014), Science and Technology Research Project of the Department of Education of Heilongjiang Province (12521401), Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities.

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